The problem of distinguishing between a centre and a focus in the space of vector fields with given Newton diagram

Abstract

We investigate the problem of distinguishing between a centre and a focus in the class of analytic vector fields with fixed Newton diagram, which satisfy certain natural conditions of general position. A method is proposed for constructing explicit expressions for the coefficients in the asymptotic representation of the monodromy transformation, known as the Dulac series. These are analogous to the Lyapunov focal quantities. These coefficients make it possible — up to an infinite-codimensional set of exceptional cases — to complete the stability analysis for a compound monodromic (that is, centre-focus) singular point. A computer-aided calculation of formulae for coefficients of the Dulac series is presented. Examples are treated of Newton diagrams with two and three edges. Bibliography: 30 titles.